A quick question about Einstein's summation

Question

I am not familiar with Einstein's summation convention. On p.5 of the book Mathematics of Classical and Quantum Physics written by Bryon and Fuller, the authors wrote: $$\mathbf x\cdot\mathbf y=x_iy_j\delta_{ij}\color{red}{=x_iy_j}=x_jy_j.$$ Why is the sum equal to $x_iy_j$? Since the expression does not contain any repeated indices, is it even a sum?

Answer 1

Where an index is repeated twice, implicitly this means to sum on that index. $x_iy_j$ does not have any implied sum ... it means take the component $x_i$ and multiply it by the component $y_j$.

This could be used to define a matrix \begin{eqnarray*} A_{i,j}=x_i y_j. \end{eqnarray*}